Question: $f(x, y) = xe^y$ $\dfrac{\partial^2 f}{\partial x^2} = $
Answer: Taking a second order partial derivative is like taking a regular second order derivative. We take the partial derivative once, then we take another partial derivative. $\dfrac{\partial^2 f}{\partial x^2} = \dfrac{\partial}{\partial x} \left[ \dfrac{\partial f}{\partial x} \right]$ Let's differentiate! $\begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left[ \dfrac{\partial}{\partial x} \left[ xe^y \right] \right] \\ \\ &= \dfrac{\partial}{\partial x} \left[ e^y \right] \\ \\ &= 0 \end{aligned}$ Therefore, $\dfrac{\partial^2 f}{\partial x^2} = 0$.